Factoring polynomials in finite fields: An application of Lang-Weil to a problem in graph theory

Abstract

Recently Chung has constructed some interesting directed graphs using finite fields, and proven non-trivial upper bounds for their diameters. We study Chung's graphs by a direct algebraic geometric method, interpreting a given bound for diameter as the statement that each of a certain collection of algebraic varieties has a rational point, and then using Lang-Weil to prove the existence of the required rational points. We are naturally led to showing that certain explicit two-parameter families of polynomials in one indeterminate have Galois group the full symmetric group